Fuzzy Systems
Mamdani-Assilian Controller
Prof. Dr. Rudolf Kruse
Christoph Doell
{kruse,doell}@iws.cs.uni-magdeburg.de
Otto-von-Guericke University of Magdeburg
Faculty of Computer Science
Department of Knowledge Processing and Language Engineering
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
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Outline
1. Motivation
Architecture of a Fuzzy Controller
Cartpole Problem
Table-based Control Function
Combination of Rules
Defuzzification
2. Example: Engine Idle Speed Control
3. Example: Automatic Gear Box
Architecture of a Fuzzy Controller
knowledge
base
fuzzification
interface
not
fuzzy
R. Kruse, C. Doell
fuzzy
measured
values
decision
logic
controlled
system
FS – Mamdani-Assilian Controller
fuzzy
defuzzification
interface
controller
output
not
fuzzy
Lecture 7
2 / 27
Example: Cartpole Problem (cont.)
X1 is partitioned into 7 fuzzy sets.
Support of fuzzy sets: intervals with length
1
4
of whole range X1 .
Similar fuzzy partitions for X2 and Y .
Next step: specify rules
if ξ1 is A(1) and . . . and ξn is A(n) then η is B,
A(1) , . . . , A(n) and B represent linguistic terms corresponding to
µ(1) , . . . , µ(n) and µ according to X1 , . . . , Xn and Y .
Let the rule base consist of k rules.
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
3 / 27
Example: Cartpole Problem (cont.)
nb
θ̇
nb
nm
ns
az
ps
pm
pb
nm
nb
nm
nm
ns
ps
ns
ns
θ
az
pb
pm
ps
az
ns
nm
nb
ps
pm
pb
ps
ps
pm
pb
pm
ns
19 rules for cartpole problem, e.g.
If θ is approximately zero and θ̇ is negative medium
then F is positive medium.
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
4 / 27
Definition of Table-based Control Function
Measurement (x1 , . . . , xn ) ∈ X1 × . . . × Xn is forwarded to decision
logic.
Consider rule
if ξ1 is A(1) and . . . and ξn is A(n) then η is B.
Decision logic computes degree to ξ1 , . . . , ξn fulfills premise of rule.
For 1 ≤ ν ≤ n, the value µ(ν) (xν ) is calculated.
o
n
Combine values conjunctively by α = min µ(1) , . . . , µ(n) .
For each rule Rr with 1 ≤ r ≤ k, compute
n
(1)
(n)
o
αr = min µi1,r (x1 ), . . . , µin,r (xn ) .
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
5 / 27
Definition of Table-based Control Function II
Output of Rr = fuzzy set of output values.
Thus “cutting off” fuzzy set µir associated with conclusion of Rr at αr .
So for input (x1 , . . . , xn ), Rr implies fuzzy set
r)
: Y → [0, 1],
µxoutput(R
1 ,...,xn
n
(n)
(1)
o
y 7→ min µi1,r (x1 ), . . . , µin,r (xn ), µir (y ) .
(1)
(n)
output(Rr )
If µi1,r (x1 ) = . . . = µin,r (xn ) = 1, then µx1 ,...,xn
(ν)
= µir .
output(Rr )
If for all ν ∈ {1, . . . , n}, µi1,r (xν ) = 0, then µx1 ,...,xn
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
= 0.
Lecture 7
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Combination of Rules
The decision logic combines the fuzzy sets from all rules.
The maximum leads to the output fuzzy set
µoutput
x1 ,...,xn : Y → [0, 1],
y 7→ max
1≤r ≤k
n
n
(1)
(n)
oo
min µi1,r (x1 ), . . . , µin,r (xn ), µir (y )
.
Then µoutput
x1 ,...,xn is passed to defuzzification interface.
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
7 / 27
Rule Evaluation
1
positive small
1
approx.
zero
1
15 25 30
45
positive medium
0.6
F
θ̇
θ
0
positive small
min
0.5
0.3
1
−8 −4 0
1
8
approx.
zero
0
min
1
3
6
9
positive medium
0.5
15 25 30
45
F
θ̇
θ
0
−8 −4 0
8
Rule evaluation for Mamdani-Assilian controller.
0
1
3
6
max
Input tuple (25, −4) leads to fuzzy output.
Crisp output is determined by defuzzification.
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
9
F
0 1
4 4.5
7.5 9
Lecture 7
8 / 27
Defuzzification
So far: mapping between each (n1 , . . . , nn ) and µoutput
x1 ,...,xn .
Output = description of output value as fuzzy set.
Defuzzification interface derives crisp value from µoutput
x1 ,...,xn .
This step is called defuzzification.
Most common methods:
• max criterion,
• mean of maxima,
• center of gravity.
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
9 / 27
The Max Criterion Method
Choose an arbitrary y ∈ Y for which µoutput
x1 ,...,xn reaches the maximum
membership value.
Advantages:
• Applicable for arbitrary fuzzy sets.
• Applicable for arbitrary domain Y (even for Y 6= IR).
Disadvantages:
•
•
•
•
Rather class of defuzzification strategies than single method.
Which value of maximum membership?
Random values and thus non-deterministic controller.
Leads to discontinuous control actions.
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
10 / 27
The Mean of Maxima (MOM) Method
Preconditions:
(i) Y is interval
output
′
(ii) YMax = {y ∈ Y | ∀y ′ ∈ Y : µoutput
x1 ,...,xn (y ) ≤ µx1 ,...,xn (y )} is
non-empty and measurable
(iii) YMax is set of all y ∈ Y such that µoutput
x1 ,...,xn is maximal
Crisp output value = mean value of YMax .
if YMax is finite:
η=
1
X
|YMax | y ∈Y
i
Max
if YMax is infinite:
yi
R
y ∈Y
η = R Max
y dy
y ∈YMax
dy
MOM can lead to discontinuous control actions.
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
11 / 27
Center of Gravity (COG) Method
Same preconditions as MOM method.
η = center of gravity/area of µoutput
x1 ,...,xn
If Y is finite, then
η=
P
η=
R
yi ∈Y yi
P
· µoutput
x1 ,...,xn (yi )
output
yi ∈Y µx1 ,...,xn (yi )
.
If Y is infinite, then
R. Kruse, C. Doell
y ∈Y y
R
· µoutput
x1 ,...,xn (y ) dy
output
y ∈Y µx1 ,...,xn (y ) dy
FS – Mamdani-Assilian Controller
.
Lecture 7
12 / 27
Center of Gravity (COG) Method
Advantages:
• Nearly always smooth behavior,
• If certain rule dominates once, not necessarily dominating again.
Disadvantage:
• No semantic justification,
• Long computation,
• Counterintuitive results possible.
Also called center of area (COA) method:
take value that splits µoutput
x1 ,...,xn into 2 equal parts.
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
13 / 27
Example
Task: compute ηCOG and ηMOM of fuzzy set shown below.
Based on finite set Y = 0, 1, . . . , 10 and infinite set Y = [0, 10].
µoutput
x1 ,...,xn (y )
1.0
ηCOG
ηMOM
0.8
0.6
0.4
0.2
y
0
0
R. Kruse, C. Doell
1
2
3
4
5
6
7
FS – Mamdani-Assilian Controller
8
9
10
Lecture 7
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Example for COG
Continuous and Discrete Output Space
ηCOG =
R 10
0
0
=
≈
ηCOG =
=
R. Kruse, C. Doell
y · µoutput
x1 ,...,xn (y ) dy
R 10
R5
0
µoutput
x1 ,...,xn (y ) dy
0.4y dy + 57 (0.2y − 0.6)y dy + 710 0.8y dy
+ 3 · 0.8
5 · 0.4 + 2 · 0.8+0.4
2
R
R
38.7333
≈ 6.917
5.6
0.4 · (0 + 1 + 2 + 3 + 4 + 5) + 0.6 · 6 + 0.8 · (7 + 8 + 9 + 10)
0.4 · 6 + 0.6 · 1 + 0.8 · 4
36.8
≈ 5.935
6.2
FS – Mamdani-Assilian Controller
Lecture 7
15 / 27
Example for MOM
Continuous and Discrete Output Space
R 10
y dy
ηMOM = R7 10
7 dy
=
25.5
50 − 24.5
=
10 − 7
3
= 8.5
7 + 8 + 9 + 10
4
34
=
4
ηMOM =
= 8.5
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
16 / 27
Problem Case for MOM and COG
1
µoutput
x1 ,...,xn
0
−2
−1
0
1
2
What would be the output of MOM or COG?
Is this desirable or not?
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
17 / 27
Outline
1. Motivation
2. Example: Engine Idle Speed Control
3. Example: Automatic Gear Box
Example: Engine Idle Speed Control
VW 2000cc 116hp Motor (Golf GTI)
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
18 / 27
Structure of the Fuzzy Controller
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
19 / 27
Deviation of the Number of Revolutions
dREV
1.00
0.75
0.50
nb
nm
ns
−50
−30
zr
ps
pm
30
50
pb
0.25
0
−70
R. Kruse, C. Doell
−10
10
FS – Mamdani-Assilian Controller
70
Lecture 7
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Gradient of the Number of Revolutions
gREV
1.00
0.75
0.50
nb
nm
ns
zr
ps
pm
pb
0.25
0
-400
R. Kruse, C. Doell
-70 -40 -30 -20
20
30
FS – Mamdani-Assilian Controller
40
70
-400
Lecture 7
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Change of Current for Auxiliary Air Regulator
dAARCUR
1.00
0.75
nm
ns
zr
ps
pm
pb
0
−25 −20 −15 −10
−5
0
5
10
15
0.50
nh
nb
ph
0.25
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
20
Lecture 7
25
22 / 27
Rule Base
If the deviation from the desired number of revolutions is negative
small and the gradient is negative medium,
then the change of the current for the auxiliary air regulation should
be positive medium.
dREV
R. Kruse, C. Doell
nb
nm
ns
az
ps
pm
pb
nb
ph
ph
pb
ps
az
az
ns
nm
pb
pb
pm
ps
az
ns
ns
ns
pb
pm
ps
az
az
ns
nm
gREV
az
pm
pm
ps
az
ns
ns
nb
FS – Mamdani-Assilian Controller
ps
pm
ps
az
az
ns
nb
nb
pm
ps
ps
az
nm
nm
nb
nb
pb
ps
az
az
ns
nb
nh
nh
Lecture 7
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Performance Characteristics
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
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Outline
1. Motivation
2. Example: Engine Idle Speed Control
3. Example: Automatic Gear Box
Example: Automatic Gear Box I
VW gear box with 2 modes (eco, sport) in series line until 1994.
Research issue since 1991: individual adaption of set points and no
additional sensors.
Idea: car “watches” driver and classifies him/her into calm, normal,
sportive (assign sport factor [0, 1]), or nervous (calm down driver).
Test car: different drivers, classification by expert (passenger).
Simultaneous measurement of 14 attributes, e.g. , speed, position of
accelerator pedal, speed of accelerator pedal, kick down, steering
wheel angle.
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
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Example: Automatic Gear Box II
Continuously Adapting Gear Shift Schedule in VW New Beetle
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
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Example: Automatic Gear Box III
Technical Details
Optimized program on Digimat:
24 byte RAM
702 byte ROM
Runtime: 80 ms
12 times per second new sport
factor is assigned.
Research topics:
When fuzzy control?
How to find fuzzy rules?
R. Kruse, C. Doell
FS – Mamdani-Assilian Controller
Lecture 7
27 / 27